Applications of Power Spectral Density [closed]

Question

I have a class covering Power Spectral Density but I have no idea why it matters. Could someone provide some examples of its use?

Thanks

Answer 1

This may sound a little glib, but actually it is not.

You know those little bars on some stereos that show you how much volume there is at each frequency band?

You're going to learn the math behind those, which also has applicability in many other places.

Ced

Answer 2

there are a couple of prerequisites.

1. first, do you know how to define a finite "power signal" vs. a finite "energy signal"?

2. here's another that's more explicit, but it's about discrete time, not continuous time.

3. and here is the beginning of a definition for Power Spectrum.

so repeating:

let $x(t)$ be a finite power signal. then, instantaneous power is:

$$ p_x(t) = |x(t)|^2 $$

the mean power is:

$$\begin{align} P_x &= \lim_{T \to \infty} \quad \frac{1}{T} \int\limits_{-\frac{T}{2}}^{+\frac{T}{2}} p_x(t) \, dt \\ \\ &= \lim_{T \to \infty} \quad \frac{1}{T} \int\limits_{-\frac{T}{2}}^{+\frac{T}{2}} |x(t)|^2 \, dt \\ \end{align}$$

now in this answer is some discussion of the sampling theorem, but i only want to use the brickwall filter results here. suppose $x(t)$ is passed through a brickwall filter having impulse response

$$ y(t) = h(t) \circledast x(t) $$

where

$$ h(t) = 2B \, \operatorname{sinc}(2B t) $$

and the frequency response is

$$ H(f) = \operatorname{rect} \left( \tfrac{f}{2B} \right) $$

so

$$ Y(f) = \begin{cases} X(f) \qquad & |f| \le B \\ 0 & |f| > B \\ \end{cases} $$

for any bandwidth $B>0$

the relation between Power Spectral Density to mean power is:

$$\begin{align} P_y &= \int\limits_{-\infty}^{\infty} S_y(f) \, df \\ &= \int\limits_{-B}^{+B} S_x(f) \, df \\ \end{align} $$

for any $B>0$.

what that means is $P_y$ is the power of $x(t)$ only at frequencies below $B$ in magnitude because all energy at frequencies above $B$ have been removed by the hypothetical brickwall low pass filter. because integral limits can add:

$$ \int\limits_{a}^{b} + \int\limits_{b}^{c} = \int\limits_{a}^{c}$$

that means that

$$ \int\limits_{f_1}^{f_2} S_x(f) \, df $$

is the power in $x(t)$ between the frequencies $f_1$ and $f_2$ (assuming $f_1 \le f_2$). suppose these two frequencies are very close. then:

$$S_x(f_0) \, \Delta f \approx \int\limits_{f_0}^{f_0+\Delta f} S_x(f) \, df $$

that means that $S_x(f_0)$ is proportional to the power around frequency $f_0$ and also proportional the tiny width of the frequency segment, $\Delta f$, around frequency $f_0$. this makes $S_x(f_0)$ a power density. it is the amount of power (in whatever units of power you make for $x(t)$) per unit frequency in the vicinity of $f_0$.